Purpose

Since it's publication in 1947, linear programming has caught on to many different parts of business and has become a key tool in making business decisions. Here, we will provide examples of linear programming used in the major functional areas of a business organization.

Finance

In the Finance world, a of the investor could be a selection problem for a mixed portfolio. In general, the number of different portfolios can be much larger than the example implies, and more and different kinds of constraints can be added with each portfolio. Another problem involves determining the funding mix for a number of products when more than one method of financing is available. The objective may be to maximize profit, where the profit for a product depends on the method of financing. For example, funding may be provided with internal funds, short-term debt, or intermediate financing (amortized loans). There may be limits on the availability of each of the funding options as well as financial constraints that require certain relationships between the funding options so they will satisfy the terms of bank loans or intermediate financing. There may also be limits of the production capacity for the products. The decision variables would be X number of units of each product to be financed by each funding option. [1]

Production and Operations Management

In the process industry, a given raw material can often be made into a wide variety of products. In the oil industry, for example, crude oil is refined into gasoline, kerosene, home-heating oil, and various grades of engine oil. Given the various profit margins for each product, we must determine the optimal quantities of each product that should be produced. There are numerous restrictions such as limits on the capacities of various refining operations, raw-material availability, demands for each product, and any government-imposed policies on the output of certain products; these restrictions become constraints in the linear programming model.[1]

Human Resources

Personnel planning problems can also be solved with linear programming. In the telephone industry, for example, demands for installer-repair personnel are seasonal. The problem is to determine the number of installer-repair personnel and line-repair personnel to have on hand each month so the total costs of hiring, layoffs, overtime, and regular-time wages are minimized. The constraints set includes limits the service demands that must be satisfied, overtime usage, union agreements, and the availability of skilled people for hire.[1]

Marketing

Linear programming can be used to determine the right mix of media exposure to use in an advertising campaign. Suppose that the available media are radio, television, and newspapers. The goal is to determine how many advertisements to place in each medium where the cost of placing an advertisement depends on the medium. Of course, we want to minimize the total cost of the advertising campaign, but subject to a series of constraints. Since each medium may provide only certain exposure of the target audience, there may be a lower bound on the total exposure from the campaign. Also, each medium may have a different efficiency rating in producing the results desired; thus, there may be a lower bound on efficiency. Plus there may be limits on the availability of each medium for an advertising platform.[1]

Distribution

Another application of linear programming is in the area of distribution. Consider a case in which there are a given number of factories that must ship goods to a given number of warehouses. Any one factory could make shipments to any number of warehouses. Given the cost to ship one unit of product from a factory to a warehouse, the problem is to determine the shipping pattern that minimizes total shipping costs. This decision is subject to the constraints that each factory cannot ship more products than it is able to produce.[1]

The uses of linear programming are not limited to these five areas but allows you to easily see why LP is so important and how it can practically be applied to many areas of decision-making.